is a computational problem. Computational problems are one of the main objects of study in theoretical computer science. The field of algorithms studies methods of solving computational problems efficiently. The complementary field of computational complexity attempts to explain why certain computational problems are intractable for computers.

A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. For example in the factoring problem, the instances are the integers n, and solutions are prime numbers p that describe nontrivial prime factors of n.

It is conventional to represent both instances and solutions by binary strings, namely elements of {0, 1}*. For example, numbers can be represented as binary strings using the binary encoding. (For readability, we identify numbers with their binary encodings in the examples below.)

A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing:

“Given a positive integer n, determine if n is prime.”

A decision problem is typically represented as the set of all instances for which the answer is yes. For example, primality testing can be represented as the infinite set

L = {2, 3, 5, 7, 11, …}

In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.

A search problem is represented as a relation over consisting of all the instance-solution pairs, called a search relation. For example, primality can be represented as the relation

R = {(4, 2), (6, 2), (6, 3), (8, 2), (8, 4), (9, 3), …}

which consist of all pairs of numbers (n, p), where p is a nontrivial prime factor of n.

A counting problem asks for the number of solutions to a given search problem. For example, the counting problem associated with primality is

“Given a positive integer n, count the number of nontrivial prime factors of n.”

A counting problem can be represented by a function f from {0, 1}* to the nonnegative integers. For a search relation R, the counting problem associated to R is the function

fR(x) = |{y: (x, y) ∈ R}|.

An optimization problem asks for finding the “best possible” solution among the set of all possible solutions to a search problem. One example is the maximum independent set problem:

In computational complexity theory, it is usually implicitly assumed that any string in {0, 1}* represents an instance of the computational problem in question. However, sometimes not all strings {0, 1}* represent valid instances, and one specifies a proper subset of {0, 1}* as the set of “valid instances”. Computational problems of this type are called promise problems.

The following is an example of a (decision) promise problem:

“Given a graph G, determine if G has an independent set of size at most 5, or every independent set in G has size at least 10.”

Here, the valid instances are those graphs whose maximum independent set size is either at most 5 or at least 10.

Decision promise problems are usually represented as pairs of disjoint subsets (Lyes, Lno) of {0, 1}*. The valid instances are those in Lyes ∪ Lno. Lyes and Lno represent the instances whose answer is yes and no, respectively.

Francisco Antonio Cerón García

The main question: What would be your next strategy step to continue developing Internet in a new radical way?
It is a way in the sense of “meta”, like Google is a “meta internet”. Do we know how to do it?
A silicon valley is essentially 90% about the people and 10% about the place. Places close to financial centres and developed cities are more likely to host the next silicon valley, but smart people can turn any place into a silicon valley if that’s what they want, even if it’s in the middle of nowhere. However, now with the Internet I believe less in silicon valleys. I mean, what’s the point of having silicon valleys when entrepreneurs and techies can network through the Net and telecommute? As everyday real life contact becomes less necessary to conduct business, we will soon start seeing the genesis of ‘virtual’ silicon valleys leveraging the power of the Internet. If Ihad to build the next silicon valley, I would start by recruiting smart people on the Internet and creating incentives for like-minded individuals and companies to participate in some sort of hub website virtual marketplace

It makes me think carefully about the next big revolutionary step on internet development. Eventually, I think that the issue that is being treated here is a key issue and it deserves a new blog to be opened for it.
Generally speaking, this is the great step that could completely change our world as far as we know it now, like when computers were created and developed or just like Google, and it is all this tiny but huge things that have been changing our way of living and the way we understand life.
This is an open question, and I want that it would be the spirit of this simple blog!
You are all invited to build the meta internet!
Then you could start thinking a lot about this issue!

Meta

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